Our sun has 8 planets orbiting as well as a number of dwarf planets. Are there any calculations that hint as to whether this number is close to some theoretical maximum value or are we simply an average solar system in this particular way?

I could imagine that if you have many planets, they will likely interact with each other. Can you calculate any theoretical value for the maximum number of planets which have long-term stable orbits around their own star?

2 Answers
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There do exist somewhat trivial configurations, which are stable in the long term and which include arbitrarily many bodies. Consider, for example, a set of $N$ circularly moving bodies of the same mass $m$, which obeys the constraint $mN\ll M$, where $M$ is the mass of the star. So long as $mN\ll M$, the bodies move dominantly in the gravitational field of the star and are hence moving stably over long term period. However, as $N$ is arbitrary, one concludes that there is no upper limit on the number of planets, provided that their total mass is small.

A more physical example would be a protoplanetary disc, or an accretion disc, which is a limit $N\rightarrow \infty$ of an arbitrary planetary system (not not necessarily circular) of a given mass. A yet more physical example is an asteroid belt, consisting of a large number of bodies on, roughly, stable orbits. Finally, during planet formation process the star goes through stages, when it is surrounded by sets of pebbles and asteroids, which keep their structure constant over a large number of orbits (roughly, of order $10^5$). And these all are real physical examples of planetary-like systems.

The answer to your question would start to alter, though, if you start imposing additional conditions apart from $N\rightarrow \infty$. For example, if you require that bodies do not collide in the long term, some of the above named systems would not work (for example, accretion disc model), but some other would (sets of concentric particles). If you additionally require that the object should obey the definition of a planet, that is have some range of masses, then interesting things will start happening when the total mass of the planets will start being comparable to the mass of the star. So the limit would certainly exist. Finally, you might be more strict about what do you really mean by stability here, and that could also have a bearing on the answer.

To summarize, unless you impose any constraints, there do exist N-body systems orbiting a star in a stable fashion and having arbitrarily large $N$.

The limit would depend on the size of the central star as well as the location and sizes of the planets in the system.

Really the limit would be the number of planets that you can fit within the area of which the orbital velocity is >0. Once you reach that distance, you can't orbit anymore. Though adding a planet would move this further out due to the added mass itself. So in theory you could keep pushing this limit and stick more planets in for forever (depending on what you consider to be a planet).

The problem comes more with having stable orbits. Each planet that you add to the system would affect the rest of the system and could cause the orbits to not be stable anymore. Also adding planets would allow more planets further out due to the additional mass but it does make figuring out if you have a stable orbit more complicated (https://en.wikipedia.org/wiki/N-body_problem).